Note on a theorem of Birch and Erdős
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910936949850112 |
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| author | Ding, Yuchen Liu, Honghu Wang, Zi |
| author_facet | Ding, Yuchen Liu, Honghu Wang, Zi |
| contents | Let $p,q>1$ be two relatively prime integers and $\mathbb{N}$ the set of nonnegative integers. Let $f_{p,q}(n)$ be the number of different expressions of $n$ written as a sum of distinct terms taken from $\{p^αq^β:α,β\in \mathbb{N}\}$. Erd\H os conjectured and then Birch proved that $f_{p,q}(n)\ge 1$ provided that $n$ is sufficiently large. In this note, for all sufficiently large number $n$ we prove $$ f_{p,q}(n)=2^{\frac{(\log n)^2}{2\log p\log q}\big(1+O(\log\log n/\log n)\big)}. $$ We also show that $\lim_{n\rightarrow\infty}f_{2,q}(n+1)/f_{2,q}(n)=1.$ Additionally, we will point out the relations between $f_{2,q}(n)$ and $m$-ary partitions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_11676 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Note on a theorem of Birch and Erdős Ding, Yuchen Liu, Honghu Wang, Zi Number Theory Let $p,q>1$ be two relatively prime integers and $\mathbb{N}$ the set of nonnegative integers. Let $f_{p,q}(n)$ be the number of different expressions of $n$ written as a sum of distinct terms taken from $\{p^αq^β:α,β\in \mathbb{N}\}$. Erd\H os conjectured and then Birch proved that $f_{p,q}(n)\ge 1$ provided that $n$ is sufficiently large. In this note, for all sufficiently large number $n$ we prove $$ f_{p,q}(n)=2^{\frac{(\log n)^2}{2\log p\log q}\big(1+O(\log\log n/\log n)\big)}. $$ We also show that $\lim_{n\rightarrow\infty}f_{2,q}(n+1)/f_{2,q}(n)=1.$ Additionally, we will point out the relations between $f_{2,q}(n)$ and $m$-ary partitions. |
| title | Note on a theorem of Birch and Erdős |
| topic | Number Theory |
| url | https://arxiv.org/abs/2503.11676 |